Abstract

The main purpose of this paper is using the properties
of Gauss sums and the mean value theorem of Dirichlet -functions
to study one kind of hybrid mean value problems involving Kloosterman sums and sums analogous to Dedekind sums and give two exact computational formulae for them.

1. Introduction

Let be a natural number and let be an integer prime to . The classical Dedekind sums
where
describe the behaviour of the logarithm of the eta-function (see [1, 2]) under modular transformations. Gandhi [3] also introduced another sum analogous to Dedekind sums as follows:
where denotes any positive even number and denotes any integer with .

About the arithmetical properties of and related sums, many authors had studied them and obtained a series of interesting results; see [1–9]. For example, the second author [7] proved the following conclusion.

Let be a positive integer with and . Then we have the asymptotic formula
where denotes the summation over all integers such that , denotes the product over all prime divisors of such that and , is the Euler function, and .

The sum is important, because it has close relations with the classical Dedekind sums . But unfortunately, so far, we knew that all results of are the properties of their own, or the relationships between and , and had nothing to do with the other arithmetic functions. If we can find some relations between and other arithmetic function, that will be very useful for further study of the properties of .

On the other hand, we introduce the classical Kloosterman sums , which are defined as follows. For any positive integer and integer ,
where denotes the solution of the congruence and .

The main purpose of this paper is using the properties of the Gauss sums and the mean square value theorem of Dirichlet -functions to study a hybrid mean value problem involving and Kloosterman sums and give two exact computational formulae for them. That is, we will prove the following.

Theorem 1. Let be an odd prime. Then one has the identity
where .

Theorem 2. Let be an odd prime; then one has the identity
where denotes the class number of the quadratic field .

2. Several Lemmas

In this section, we will give several lemmas, which are necessary in the proof of our theorems. Hereinafter, we will use many properties of Gauss sums, all of which can be found in [12], so they will not be repeated here. First we have the following.

Lemma 3. Let be an odd prime; then one has the identity

Proof. It is clear that if pass through a complete residue system , then also pass through a complete residue system . So for any nonprincipal character , from the properties of Gauss sums (see Theorem 8.9 of [12])
we have the identity
This proves Lemma 3.

Lemma 4. Let be an integer; then for any integer with , one has the identity
where denotes the Dirichlet -function corresponding to character .

Lemma 5. Let be an odd prime. Then for any odd number with , one has the identity
where satisfies the congruence .

Proof. Note that the divisors of are , and . So from Lemma 4 and the definition of and we have
where denotes the principal character .From the Euler infinite product formula (see Theorem 11.6 of [12]) we have,
where denotes the product over all primes .From Lemma 4 we also have the identity
Note that is an odd number; combining (14), (15), and (16) we have the identity
This proves Lemma 5.

Lemma 6. Let be an odd prime. Then one has the identities(A) (B)

Proof. From the definition of Dedekind sums we have
If , then, from (20) and noting that the reciprocity theorem of Dedekind sums (see [5]), we have the computational formula
Now taking in (21), from (16) we may immediately deduce the identity
Taking in (21), from (16) we can also deduce the identity
Now Lemma 6 follows from (22) and (23).

3. Proof of the Theorems

In this section, we will complete the proof of our theorems. First we prove Theorem 1. Note that if is a nonprincipal character , then and
From (24) and Lemmas 4, 5, and 6 we have
This proves Theorem 1.

Now we prove Theorem 2. If , then from Lemmas 3, 5, and 6 we have
If , then note that the Legendre symbol , (see Dirichlet's class number formula, Chapter 6 of [13]), and
so from Lemmas 3, 5, and 6 we have
Note that if ; and if , from (28) we may immediately deduce
Now Theorem 2 follows from (26) and (29).

This completes the proofs of all results.

Acknowledgments

The authors would like to thank the referee for their very helpful and detailed comments, which have significantly improved the presentation of this paper. This work is supported by the NSF (11071194) of China.